Hypergraceful complete graphs.
Siddani Bhaskara Rao, B. Devadas Acharya, Tarkeshwar Singh, Mukti Acharya · Australas. J Comb. · 2010
In this paper we prove that no signed graph on the complete graph Kp, p ≥ 6, is graceful, and we also give a characterization of graceful signed graphs on Kp, p ≤ 5. This implies that there is no subset A of cardinality p ≥ 6 from the set {0, 1, . . . , (p 2 ) − n}, n ≤ 1 2 ( p 2 ) , such that each element of the set {1, 2, . . . , n} occurs exactly twice and each element of the remaining set {n + 1, n + 2, . . . , (p 2 ) − n} occurs exactly once as an absolute value of a pair of distinct elements of A. Also, all such subsets of cardinality p ≤ 5 are determined. 6 RAO, ACHARYA, SINGH AND ACHARYA